@vonbrand, $AC^0$ is the class of constant depth circuits containing and/or gates of unbounded fan-in. That is, each gate in a circuit is either an "and" or an "or" gate, and allow an unbounded number of inputs coming in.
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Nicholas MancusoFeb 12 '13 at 21:20

1 Answer
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Every unary language is in nonuniform $AC^0$; for example, the halting problem expressed in unary.

Addition can be implemented in $AC^0$ with a carry-lookahead adder. Here the input is $2n$ bits representing two numbers, and the output contains $n+1$ wires (equivalently, each output bit can be realized in $AC^0$)

Multiplexing: $\{w x: |w|=2^n, |x|=n, w[x] = 1\}$ is in $AC^0$.

A multiplexer is a function on $2^n+n$ variables which outputs the value of one of $2^n$ variables, where the index is determined by the $n$ variables. (The same holds if the index is written in unary.)

Computation of 3SAT formulas is in $AC^0$.

The input consists of $n$ variables, followed by some clauses, each one contains three literals, where each literal is an index of the variable (unary or binary, does not matter) and a bit indicating possible negation. You can evaluate the literals with multiplexers and then add a layer of ORs and then a big AND on top.

$AC^0$ does not contain majority, but it contains approximate majority: a function that is equal to majority if the output is $\geq \frac{1}{2}+ \varepsilon$ zeroes or ones. See "Approximate Counting with Uniform Constant-Depth Circuits" by Ajtai.

$AC^0$ is closed under logical operations, concatenation and composition, so you can combine above examples. Now you should feel some respect for $Parity \notin AC^0$ and other circuit lower bounds!

Do you have some references to this? Especially that unary halting problem is in $AC^0$. Since $AC^0 \subseteq AC = NC \subseteq P$, I don't get this (it's late where I am, that might be my excuse).
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Pål GDFeb 13 '13 at 0:19